Lesson 84 Identifying Quadratic functions Quadratic Function A

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Lesson 84 Identifying Quadratic functions

Lesson 84 Identifying Quadratic functions

Quadratic Function • A function pairs each value in the domain with exactly one

Quadratic Function • A function pairs each value in the domain with exactly one value in the range. • A quadratic function is a function that can be written in the form • f(x) = ax 2 + bx + c, • Quadratic term linear term constant term • where a is not equal to 0. • All quadratic functions consist of a polynomial with a degree of 2 ( the degree of a polynomial is the same as the degree of its greatest degree term)

degrees • Polynomial • 3 x+2 • x 2+4 x-5 • 2 x 3

degrees • Polynomial • 3 x+2 • x 2+4 x-5 • 2 x 3 -x 2+1 degree 1 2 3 name linear quadratic cubic

Standard form of a quadratic function • f(x) = 2 ax + bx +

Standard form of a quadratic function • f(x) = 2 ax + bx + c • a, b, c are real numbers and a is not 0.

Identifying quadratic functions • Determine whether each is a quadratic function ( can it

Identifying quadratic functions • Determine whether each is a quadratic function ( can it be written in the standard form of a quadratic function) • y+7 x=4 x 2 -6 • y= 5+ 2 x • -2 x 3 + y = -5 x 3 + x 2 • y-15 x 2=3+2 x • y-62=3 -2 x • 9 x 4 -9=y+8 x

Graphing quadratic functions • The graph of f(x) = x 2 is known as

Graphing quadratic functions • The graph of f(x) = x 2 is known as the quadratic parent function. • Make a table of values and graph. The graph is a parabola.

Graphing using a table • Graph f(x) = -3 x 2 • use the

Graphing using a table • Graph f(x) = -3 x 2 • use the points , -2, -1, 0, 1, 2 for x and find y- then plot the points with a smooth curve. • Graph f(x) = 5 x 2 - 3

Direction of a parabola • For a quadratic function in standard form, • y

Direction of a parabola • For a quadratic function in standard form, • y = ax 2 + bx + c • If a < 0, the parabola opens downward • If a> 0, the parabola opens upward

Determining the direction of a parabola • Does it open upward or downward? •

Determining the direction of a parabola • Does it open upward or downward? • f(x) = 3 x 2 + 8 2 • f(x) = 3 x-x + 5 2 • f(x) = 3 x - 4 x + 7 • f(x) = 15 -10 x - x 2

Lesson 89 • Identifying characteristics of quadratic functions

Lesson 89 • Identifying characteristics of quadratic functions

Quadratic terms • The vertex of a parabola is the highest or lowest point

Quadratic terms • The vertex of a parabola is the highest or lowest point on a parabola- it is the parabola's "turning point" • The minimum of a function is the least possible value of a function. • The maximum of a function is the highest possible value of a function. • It is the y-value of the lowest or highest point on the graph of the function. • On a parabola, the minimum or maximum is the y-coordinate of the vertex.

Identifying the vertex and the maximum or minimum • Give the coordinates of the

Identifying the vertex and the maximum or minimum • Give the coordinates of the vertex and the maximum or minimum value and the domain and range of the function

Intercepts, zeros, and roots • An x-intercept of a function is the xcoordinate of

Intercepts, zeros, and roots • An x-intercept of a function is the xcoordinate of the point where the graph intersects the x-axis. • A zero of a function is the value of x that makes f(x) = 0, or y=0 • Because y=0 for every value of x on the x-axis, the zeros of a function are the same as the x-intercepts

Finding zeros from the graph

Finding zeros from the graph

Axis of symmetry • An axis of symmetry is a line that divides a

Axis of symmetry • An axis of symmetry is a line that divides a figure or graph into two mirrorimage halves • All parabolas have an axis of symmetry that passes through the vertex. • The equation of the axis of symmetry includes the x-coordinate of the vertex and it is the average of the zeros.

Axis of symmetry formula • the axis of symmetry for the graph of a

Axis of symmetry formula • the axis of symmetry for the graph of a quadratic equation • y = ax 2 +bx + c is • x= -b 2 a

Finding axis of symmetry using formula • Find the axis of symmetry 2 •

Finding axis of symmetry using formula • Find the axis of symmetry 2 • y= x + 6 x + 5 • x = -b = -6 = -3 • 2 a 2(1)

Find axis of symmetry • y= -4 x +3 2 • y= 3 x

Find axis of symmetry • y= -4 x +3 2 • y= 3 x + 5 2 -x

Lab 8 • Characteristics of parabolas

Lab 8 • Characteristics of parabolas

Using a calculator • You can use a graphing calculator to compute approximate values

Using a calculator • You can use a graphing calculator to compute approximate values of any x-intercepts and the maximum and minimum of the parabola

Find x-intercepts and minimum or maximum • • y = -2 x 2+4 x-2

Find x-intercepts and minimum or maximum • • y = -2 x 2+4 x-2 Enter the equation into Y= Press ZOOM 6: standard to graph the equation Press TRACE and then use the right and left arrow keys to move along the curve to the approximate x-intercept - this will give you the approximate values at the bottom of the screen. Press 2 nd TRACE (which is CALC) and select 2: zero Trace along the curve to a point left of the x-intercept and press ENTER. Trace along the curve to the right of the x- intercept and press ENTER. Press ENTER again-this will give you the exact value of the zero

To find minimum or maximum • Enter the equation into the Y= editor •

To find minimum or maximum • Enter the equation into the Y= editor • Press ZOOM: Standard to graph • Press 2 nd TRACE and select 3: minimum or 4: : maximum • Follow same procedure as for zeros.